Compound-angle formulae and geometric proofs
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| compound angle/ˈkɒmpaʊnd ˈæŋɡl/ | góc tổng hợp |
Can its length explain why cosine of a difference is not the difference of two cosines?
- Two unit-radius points define a chord.
- This lesson studies compound angle 复合角: An angle expressed as the sum or difference of two angles.
The method
- sin(A±B)=sinA cosB±cosA sinB. cos(A±B)=cosA cosB∓sinA sinB: the cosine sign changes. tan(A±B)=(tanA±tanB)/(1∓tanA tanB) where the tangents and denominator are defined.
- Derive tangent by dividing the corresponding sine and cosine formulas; do not extend a quotient through a zero denominator.
Why is cos(A+B)=cosA cosB - sinA sinB while sin(A+B) carries a plus?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Check with A=B=45°: cos 90°=0 and cosAcosB-sinAsinB=1/2-1/2=0, while a plus would give 1. The cosine sign reversal is exactly what the compound geometry produces.
A worked case
Let P=(cosA,sinA) and Q=(cosB,sinB) on the unit circle. Coordinate distance gives PQ²=2−2(cosA cosB+sinA sinB). The cosine rule on the two unit radii gives PQ²=2−2cos(A−B), hence the cosine difference formula. Replace B by −B, using cosine even/sine odd, to obtain cosine addition. Apply cosine difference to (π/2−A) and B to obtain sin(A+B); then change B’s sign for sine subtraction. Exact evaluations: sin75°=sin(45°+30°)=(√6+√2)/4; cos75°=(√6−√2)/4; sin15°=(√6−√2)/4, cos15°=(√6+√2)/4 and tan15°=2−√3.
Compound-angle formulae and geometric proofs
sin(A±B)=sinA cosB±cosA sinB
Explain how the angle formula follows from the geometric or coefficient conditions.
Find sin(π/12) as a decimal. Give it to 3 significant figures.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Use sin(π/4−π/6)=(√6−√2)/4≈0.258819.
Where it goes wrong
- sin(A+B) is not sinA+sinB.
- The cosine rule chord uses the smaller included angle when necessary, but its cosine is still cos(A−B).
- Coordinate distance and the resulting identities hold for all signed angles; the initial diagram is one illustrative case. tan(45°+45°) is undefined because 1−tan45°tan45°=0.
Not true: Cosine of a sum equals the sum of the two cosine values. cos(A+B)=cos A cos B−sin A sin B, which is not cos A+cos B.
Find cos(π/12) as a decimal. Give it to 3 significant figures.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Use cos(π/4−π/6)=(√6+√2)/4≈0.965926.
Cosine of a sum equals the sum of the two cosine values.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
cos(A+B)=cos A cos B−sin A sin B, which is not cos A+cos B.
Applying it
- Choose familiar angles whose sum or difference is the target.
- Keep exact roots and rationalise a tangent quotient if needed.
- In a geometric proof, state both expressions for the same chord rather than assuming the desired identity.
- Check the denominator before applying a tangent addition/subtraction formula.
Find tan(π/12) as a decimal. Give it to 3 significant figures.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Use tan(π/4−π/6)=2−√3≈0.267949.
Match each exact value to its compound-angle computation.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The special-triangle values feed the compound formulas to produce exact values at 15° and 75° without a calculator.
In your course
- AQA A-level Mathematics (7357), specification section E.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Compound angle: An angle expressed as the sum or difference of two angles.
- sin(A±B)=sinA cosB±cosA sinB. cos(A±B)=cosA cosB∓sinA sinB: the cosine sign changes. tan(A±B)=(tanA±tanB)/(1∓tanA tanB) where the tangents and denominator are defined.
- Watch for: sin(A+B) is not sinA+sinB.